Dual of a differential graded module Let $(M, d_M)$ be a differential $\mathbb{Z}$-graded module over a differential graded algebra (over a field) $(A, d_A)$.
I am wondering if there is a canonical way of looking at the dual $\hom_A(M,A)$  as a differential graded module over $A$. ? I am mainly concerned on the grading that the dual carries.
 A: 1) Regarding gradings, you want $\text{Hom}_k(M, N)$ to be a chain complex, it should be graded so that $f: C \to D$ has $|c| + |f| = |f(c)|$; that is, the homological degree of $f$ is how much it increases the degree of an element of $C$.
The differential is defined as $d_Nf - (-1)^{|f|} f d_M$. Explicitly, if $c \in C$, $$(df)(c) = d_N f(c) - (-1)^{|f|} f(d_M c).$$ 
This is the sign convention used in Dold's book on algebraic topology and Tyler Lawson's note on signs. It is the convention so that the pairing $\text{Hom}_k(M, N) \otimes M \to N$, given by $f \otimes m \mapsto f(m)$, is a chain map. 
If $M$ and $N$ are left $A$-modules, $\text{Hom}_A(M, N)$ is a subcomplex of $\text{Hom}_k(M, N)$, so the above discussion applies. 
2) If $M$ and $N$ are only left $A$-modules, $\text{Hom}_A(M, N)$ has no extra structure. If $N$ is a bimodule (it carries both a left and right action of $A$), then the right action survives; so $\text{Hom}_A(M, N)$ has a right $A$-module structure, via $$(\psi \cdot a)(m) = (-1)^{|a| \cdot |m|} \psi(m) a,$$ though I wouldn't bet my life on the sign; I haven't checked it. In particular, this is true of $\text{Hom}_A(M, A)$. 
